Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.[1] Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.

Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmeticformulas. Volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shape's boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using three-dimensional techniques such as the Body Volume Index. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.

The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the combined volume is not additive.[2]

Any unit of length gives a corresponding unit of volume, namely the volume of a cube whose side has the given length. For example, a cubic centimetre (cm3) would be the volume of a cube whose sides are one centimetre (1 cm) in length.

Related terms

Capacity is defined by the Oxford English Dictionary as "the measure applied to the content of a vessel, and to liquids, grain, or the like, which take the shape of that which holds them".[4] (The word capacity has other unrelated meanings, as in e.g. capacity management.) Capacity is not identical in meaning to volume, though closely related; the capacity of a container is always the volume in its interior. Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length. In SI the units of volume and capacity are closely related: one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial; the capacity of a vehicle's fuel tank is rarely stated in cubic feet, for example, but in gallons (a gallon fills a volume of 0.1605 cu ft).

h = any dimension of the figure,A(h) = area of the cross-sections perpendicular to h described as a function of the position along h. a and b are the limits of integration for the volumetric sweep.
(This will work for any figure if its cross-sectional area can be determined from h).

and are functions expressing the outer and inner radii of the function, respectively.

Volume ratios for a cone, sphere and cylinder of the same radius and height

A cone, sphere and cylinder of radius r and height h

The above formulas can be used to show that the volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3, as follows.

Let the radius be r and the height be h (which is 2r for the sphere), then the volume of cone is

the volume of the sphere is

while the volume of the cylinder is

The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes.[7]

Volume formula derivations

Sphere

The volume of a sphere is the integral of an infinite number of infinitesimally small circular disks of thickness dx. The calculation for the volume of a sphere with center 0 and radius r is as follows.

The surface area of the circular disk is .

The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is

or

where y or z can be taken to represent the radius of a disk at a particular x value.

Using y as the disk radius, the volume of the sphere can be calculated as

Now

Combining yields

This formula can be derived more quickly using the formula for the sphere's surface area, which is . The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to

=

Cone

The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies to cones as well.

However, using calculus, the volume of a cone is the integral of an infinite number of infinitesimally thin circular disks of thickness dx. The calculation for the volume of a cone of height h, whose base is centered at (0,0,0) with radius r, is as follows.

The radius of each circular disk is r if x = 0 and 0 if x = h, and varying linearly in between—that is,

The surface area of the circular disk is then

The volume of the cone can then be calculated as

and after extraction of the constants:

Integrating gives us

Volume in differential geometry

In differential geometry, a branch of mathematics, a volume form on a differentiable manifold is a differential form of top degree (i.e. whose degree is equal to the dimension of the manifold) that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. Integrating the volume form gives the volume of the manifold according to that form.

References

↑One litre of sugar (about 970 grams) can dissolve in 0.6 litres of hot water, producing a total volume of less than one litre. "Solubility". Retrieved 2010-05-01. Up to 1800 grams of sucrose can dissolve in a liter of water.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>